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Erschienen in:
2012 | OriginalPaper | Buchkapitel
1. Basic (Elementary) Inequalities and Their Application
verfasst von : Dipl. Math. Zdravko Cvetkovski
Erschienen in: Inequalities
Verlag: Springer Berlin Heidelberg
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Abstract
There are many trivial facts which are the basis for proving inequalities. Some of them are as follows: These properties are obvious and simple, but are a powerful tool in proving inequalities, particularly Property 5, which can be used in many cases.
1.
If x≥y and y≥z then x≥z, for any x,y,z∈ℝ.
2.
If x≥y and a≥b then x+a≥y+b, for any x,y,a,b∈ℝ.
3.
If x≥y then x+z≥y+z, for any x,y,z∈ℝ.
4.
If x≥y and a≥b then xa≥yb, for any x,y∈ℝ+ or a,b∈ℝ+.
5.
If x∈ℝ then x
2≥0, with equality if and only if x=0. More generally, for A
i
∈ℝ+ and x
i
∈ℝ,i=1,2,…,n holds \(A_{1} x_{1}^{2} + A_{2} x_{2}^{2} +\cdots+ A_{n} x_{n}^{2} \geq 0\), with equality if and only if x
1=x
2=⋯=x
n
=0.